# SEP-108 (2001)

## Download

Book (pdf)

## Angle-gathers

### Amplitude-preserved wave-equation migration

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**Sava P. and Biondi B.**

We analyze the amplitude variation as a function of reflection angle (AVA) for angle-domain common image gathers (ADCIG) produced via wave-equation migration. Straightforward implementations of the two main ADCIG methods lead to contradictory, thus inaccurate, amplitude responses. The amplitude inaccuracy is related to the fact that downward-continuation migration is the adjoint of upward-continuation modeling, but it is only a poor approximation of its inverse. We derive the frequency-wavenumber domain diagonal weighting operators that make migration a good approximation to the inverse of modeling. With these weights, both ADCIG methods produce consistent results. The main applications that follow from this paper are true-amplitude migration and pseudo-unitary modeling/migration, usable for iterative inversion. The two most important factors that degrade the accuracy of wave-equation ADCIGs are the limited sampling and offset range, combined with the band-limited nature of seismic data.

Offset and angle domain common-image gathers for shot-profile migration

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**Rickett J. and Sava P.**

In order to estimate elastic parameters of the subsurface, geophysicists need reliable information about angle-dependent reflectivity. In this paper, we describe how to image non-zero offsets during shot-profile migration so that they can be mapped to the angle domain with Sava and Fomel's 2000 transformation. CIGs also contain information about how well focused events are at depth, and so provide a natural domain for migration-focusing velocity analysis.

*PS*-wave polarity reversal in angle domain common-image gathers

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**Rosales D. and Rickett J.**

The change in the reflection polarity at normal incidence is a fundamental feature of converted-wave seismology due to the vector nature of the displacement field. The conventional way of dealing with this feature is to reverse the polarity of data recorded at negative offsets. However, this approach fails in presence of complex geology. To solve this problem we propose operating the polarity flip in the angle domain. We show that this method correctly handle the polarity reversal after prestack migration for arbitrarily complex earth models.

Amplitude analysis in the angle domain

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**Gratwick D.**

This paper discusses amplitude vs. angle (AVA) analysis using image gathers generated from a wave-equation migration algorithm. An AVA cross-plot muting algorithm is used to highlight parts of an image corresponding to a Class III, low impedance, AVO sand. Processing to eliminate surface multiples is used on the synthetic data, thereby enhancing reflectors. Results show that our AVA muting algorithm is effective for both a synthetic and a real dataset.

The accuracy of wave-equation migration amplitudes

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**Gratwick D.**

When migrating seismic data for the purpose of reservoir characterization, it is necessary to use a migration algorithm that preserves relative amplitude trends Scheriff (1995). In the industry, this is usually attained using Kirchhoff methods with asymptotic Green's functions Biondi (2000). This method is useful in many geologic settings, but when a complex velocity Earth introduces more complex wave propagation phenomena, ``wave-equation'' migration (WEM) based on downward continuation becomes more attractive Prucha et al. (1999). ...

Amplitude behavior in angle gathers

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**Mora C. B.**

In a previous report, Sava and Fomel (2000) presented a method for computing angle-domain common-image gathers by a radial-trace transform in the Fourier domain. The method converts offset-domain common-image gathers, which are computed using 2-D prestack wave-equation migration Prucha et al. (1999) into true reflection angle-domain common-image gathers. ...

## Imaging

### Model-space vs. data-space normalization for recursive depth migration

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**Rickett J.**

Illumination problems caused by finite-recording aperture and lateral velocity lensing can lead to amplitude fluctuations in migrated images. I calculate both model and data-space weighting functions that compensate for these illumination problems in recursive depth migration results based on downward-contination. These weighting functions can either be applied directly with migration to mitigate the effects of poor subsurface illumination, or used as preconditioning operators in iterative least-squares (*L*2) migrations. Computational shortcuts allow the weighting functions to be computed at about the cost of a single migration. Results indicate that model-space normalization can significantly reduce amplitude fluctuations due to illumination problems. However, for the examples presented here, data-space normalization proved susceptible to coherent noise contamination.

Imaging under salt edges: A regularized least-squares inversion scheme

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**Prucha M. L., Clapp R. G., and Biondi B. L.**

We introduce a method for improving the image in areas of poor illumination using least-squares inversion regularized with dip penalty filters in one and two dimensions. The use of these filters helps to emphasize the weak energy that exists in poorly illuminated areas, and fills-in gaps by assuming lateral continuity along the reflection-angle axis and/or the midpoint axes. We tested our regularized inversion method on synthetic and real data. The inversion employing one-dimensional filters along the reflection-angle axis generated prestack images significantly better than the images obtained by simple migration and unregularized inversion. The inversion employing two-dimensional filters reduced the frequency of the image but also increased reflectors' continuity and reduced noise.

Narrow-azimuth migration: Analysis and tests in vertically layered media

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**Biondi B.**

Analysis of common-azimuth migration for vertically layered media shows that downward continuing the data in a narrow strip around the zero crossline offset should yield a kinematically correct migration scheme. I introduce two migration methods that exploit the common-azimuth equations to define an optimal range of crossline-offset wavenumbers and thus to minimize the number of crossline offsets that are necessary to sample adequately the crossline-offset dips. Tests on synthetic data generated assuming a vertically layered medium confirm the theoretical analysis and suggest further testing on data sets with complex velocity functions.

Analysis of the damping factor in phase-shift migration

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**Rosales D.**

Cosmetic processes, the use of different parameters in standard seismic data processing in order to improve the appearance of the data, are generally not considered in the mathematical formulation of migration algorithms, even though they are physically and mathematically related to the wave propagation process. The inclusion of causality and viscosity in phase-shift migration as a damping factor will take care of these ``superficial'' features and numerical instability due to evanescent energy.

Acoustic daylight imaging: Introduction to the underlying concept: A prospect for the instrumented oil field

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**Claerbout J.**

Why and how it is that the autocorrelation of natural noise gives us a reflection seismogram.

Preliminary results from a small-scale 3-D passive seismic study in Long Beach, CA

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**Kerr B. and Rickett J.**

While helioseismologists routinely crosscorrelate stochastic acoustic noise to produce time-distance curves Duvall et al. (1993) that look like active-source seismograms, terrestrial geophysicists have had less success. Baskir and Weller (1975) describe the first published attempt to use passive seismic energy to image subsurface reflectivity. They briefly describe crosscorrelating long seismic records to produce correlograms that could be processed, stacked and displayed as conventional seismic ...

## Inversion and velocities

### Multiple realizations: Model variance and data uncertainty

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**Clapp R. G.**

Geophysicists typically produce a single model, without addressing the issue of model variability. By adding random noise to the model regularization goal, multiple equi-probable models can be generated that honor some *a priori* estimate of the model's second-order statistics. By adding random noise to the data, colored by the data's covariance, equi-probable models can be generated that give an estimate of model uncertainty resulting from data uncertainity. The methodology is applied to a simple velocity inversion problem with encouraging results.

A least-squares approach for estimating integrated velocity models from multiple data types

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**Brown M. and Clapp R. G.**

Many exploration and drilling applications would benefit from a robust method of integrating vertical seismic profile (VSP) and seismic data to estimate interval velocity. In practice, both VSP and seismic data contain random and correlated errors, and integration methods which fail to account for both types of error encounter problems. We present a nonlinear, tomography-like least-squares algorithm for simultaneously estimating an interval velocity from VSP and seismic data. On each nonlinear iteration of our method, we estimate the optimal shift between the VSP and seismic data and subtract the shift from the seismic data. In tests, our algorithm is able to resolve an additive seismic depth error, caused by a positive velocity perturbation, even when random errors are added to both seismic and VSP data.

A differential scheme for elastic properties of rocks with dry or saturated cracks

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**Berryman J. G., Pride S. R., and Wang H. F.**

Differential effective medium (DEM) theory is applied to the problem of estimating physical properties of elastic media with penny-shaped cracks, filled either with air or liquid. These cracks are assumed to be randomly oriented. It is known that such a model captures many of the essential physical features of fluid-saturated or partially saturated rocks. By making the assumption that the changes in certain factors depending only on Poisson's ratio do not strongly affect the results, it is possible to decouple the equations for bulk (*K*) and shear (*G*) modulus, and then integrate them analytically. The validity of this assumption is then tested by integrating the full DEM equations numerically. The analytical and numerical curves for both *K* and *G* are in very good agreement over the whole porosity range of interest. Justification of the Poisson's ratio approximation is also provided directly by the theory, which shows that, as porosity tends to 100%, Poisson's ratio tends towards small positive values for dry, cracked porous media and tends to one-half for liquid saturated samples. A rigorous stable fixed point is obtained for Poisson's ratio, , of dry porous media, where the location of this fixed point depends only on the shape of the voids being added. Fixed points occur at for spheres, and for cracks, being the aspect ratio of penny-shaped cracks. Results for the elastic constants are then compared and contrasted with results predicted by Gassmann's equations and with results of Mavko and Jizba, for both granite-like and sandstone-like examples. Gassmann's equations do not predict the observed liquid dependence of the shear modulus *G* at all. Mavko and Jizba predict the observed dependence of shear modulus on liquid bulk modulus for small crack porosity, but fail to predict the observed behavior at higher porosities. In contrast, the analytical approximations derived here give very satisfactory agreement in all cases for both *K* and *G*.

oclib: An out-of-core optimization library

(ps.gz 39K) (pdf 64K) (src 49K)

**Sava P.**

This paper introduces a Fortran90 out-of-core optimization library designed for large-scale problems. The library is centered around the filtering operators and gradient solvers currently in use at SEP.

## s=d-n

### Coherent noise attenuation: A synthetic and field example

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**Guitton A.**

Noise attenuation using either a filtering or a subtraction scheme is achieved as long as the prediction error filter (PEF), which (1) filters the coherent noise in the first method and (2) models the noise in the second one, can be accurately estimated. If a noise model is not known in advance, I propose estimating the PEF from the residual of a previous inverse problem. At this stage, the filtering and subtraction method give similar results on both synthetic and real data. However the subtraction method can more completely separate the noise and signal when both are correlated.

A pattern-based technique for ground-roll and multiple attenuation

(ps.gz 1964K) (pdf 1794K) (src 5726K)

**Guitton A., Brown M., Rickett J., and Clapp R.**

We present a pattern-based method that separates coherent noise from signal. This method finds its mathematical foundation in the work conducted by Nemeth (1996) on coherent noise attenuation by least-squares migration. We show that a similar inverse problem can be formulated to attenuate coherent noise in seismic data. In this paper, we use deconvolution with prediction error filters to model the signal and noise vectors in a least-squares sense. This new formulation of the noise separation problem has been tested on 2-D real data for ground-roll and multiple attenuations. So far, it achieves similar results to the approach used by Brown and Clapp (2000) and Clapp and Brown (2000). However, we show that the main strength of this new method is its ability to incorporate regularization in the inverse problem in order to decrease the correlation effects between noise and signal.

Adaptive multiple subtraction with non-stationary helical shaping filters

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**Rickett J., Guitton A., and Gratwick D.**

We suppress surface-related multiples with a smart adaptive least-squares subtraction scheme in the time-space domain after modeling multiples with a fast but approximate modeling algorithm. The subtraction scheme is based on using a linear solver to estimate a damped non-stationary shaping filter. We improve convergence by preconditioning with a space-domain helical roughening filter.

Solutions to data and operator aliasing with the parabolic radon transform

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**Guitton A.**

Focusing in the radon domain can be affected by data and operator aliasing. Antialiasing conditions can be imposed on the parabolic radon transform (PRT) operator by dip limiting the summation path. These dip limits in time translate into frequency limits in the Fourier domain. Consequently, antialiasing the PRT enables better focusing in the radon domain. If the radon domain is computed *via* inverse theory, a regularization term in either the time or frequency domain can reduce data aliasing effects. The frequency domain regularization has the advantage of being noniterative, but needs to be applied in patches in order to improve focusing.

Multiple suppression with land data: A case study

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**Alvarez G.**

Some of the most important methods for multiple suppression are based on the *moveout* difference between the hyperbolas corresponding to primary and multiple reflections in a CMP gather. This moveout difference is exploited by means of the Parabolic Radon Transform. In this case study I review the methodology and show the result of its application to a 2-D land seismic line are. Of particular importance are the results that show that without the suppression of multiples a distorted image is obtained of the Paleozoic and its stratigraphic terminations against basement, which constitute the exploratory objective in the area. This is partly due to the improved stacking velocities afforded by the suppression of the multiples.

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